This invention relates generally to methods, apparatus, and software for iterative image reconstruction and, more particularly, to accurate geometric forward modeling.
In Computed Tomography (CT) diagnostic imaging, the evolution of the scanning geometry into three-dimensional space as the detector size increases and the scanning trajectory evolves is creating a new category of issues for image reconstruction algorithms. In particular, the increasing number of slices on the detector creates significant deviation from the conventional two-dimensional representation of conventional reconstruction techniques, such as Filtered Back-Projection (FBP). Accounting for the conic nature of the x-ray beam becomes desirable to eliminate image artifacts. Among other techniques that are currently being studied to address the issue, Iterative Reconstruction (IR) is a candidate with excellent promise. It allows inclusion of the exact geometry of the scanning trajectory though accurate modeling.
Iterative Reconstruction relies on successive operations of forward and backprojection to obtain the convergence of a derived optimization criterion, which is a measure of the difference between the true measurements and the forward projection of the estimated reconstructions. Let x be the discrete vector of three-dimensional reconstructed space. x's elements represent the unknown densities of the elements of space forming the 3-D volume and are the object of the reconstruction. In addition, let y be the discrete vector of measurements along a number of projection directions. y's elements represent the line integrals through the imaged object for a variety of positions and projection angles that follow the acquisition trajectory and let F(x) be the expected values of the sinogram when the 3-D cross section being reconstructed is assumed to be x. Importantly, the model F(x) includes the precise geometry of the scan pattern and the source/detector structure, so it can directly account for the scan measurements. The difference between the measurements y and their expected values is commonly referred to as “noise,” and may be incorporated into the model in the equation y=F(x)+n, where n represents the noise vector. Then, the reconstruction problem may be transformed into the following optimization problem
                              x          ^                =                  arg          ⁢                                          ⁢                                    min              x                        ⁢                          {                                                                    ∑                                          i                      =                      0                                        M                                    ⁢                                                            D                      i                                        ⁡                                          (                                                                        y                          i                                                ,                                                                              F                            i                                                    ⁡                                                      (                            x                            )                                                                                              )                                                                      +                                  U                  ⁡                                      (                    x                    )                                                              }                                                          (        1        )            
where the functional Di penalizes the distance between measurement i and the corresponding simulated i-th forward projection of x, U is a scalar valued regularization term which penalizes local differences between voxel elements, and F is a transformation of the image space x in a manner similar to the CT scanning system. A common embodiment of (1) takes the form
                              x          ^                =                  arg          ⁢                                          ⁢                                    min              x                        ⁢                          {                                                                    ∑                                          i                      =                      0                                        M                                    ⁢                                                            w                      i                                        ⁢                                                                                                                                                y                            i                                                    -                                                                                    F                              i                                                        ⁡                                                          (                              x                              )                                                                                                                                                  2                                                                      +                                  U                  ⁡                                      (                    x                    )                                                              }                                                          (        2        )            where wi is a constant which weights the contribution of measurement i to the objective function. Frequently, a linear model of the form F(x)=Ax is used, linearizing the relation between x and y with a matrix A.
The iterative reconstruction (IR) algorithm works by finding perturbations of the image space x that help to minimize the quantity in (1) or (2), which is a measure of the discrepancy between the actual measurements y and the estimated forward projection F(x) of the image space. It would be desirable to provide improvements to the known IR algorithms.